Cauchy problem for non-linear systems of equations in the critical case
- M.V. Lomonosov Moscow State University, Moscow (Russian Federation)
The large-time asymptotic behaviour is studied for a system of non-linear evolution dissipative equations ; u{sub t}+N(u,u)+Lu=0, x element of R{sup n}, t>0; u(0,x)=u-tilde(x), x element of R{sup n}; where L is a linear pseudodifferential operator Lu=F-bar{sub {xi}}{sub {yields}}{sub x}(L({xi}u-bar({xi})) and the non-linearity N is a quadratic pseudodifferential operator N(u,u{sub =}F-bar{sub {xi}}{sub {yields}}{sub x}{integral}{sub R{sup n}}A{sup kl}(t,{xi},y)u-bar{sub k}(t,{xi}-y)u-bar{sub l}(t,y)dy, where u-bar{identical_to}F{sub x{yields}}{sub {xi}}u is the Fourier transform. Under the assumptions that the initial data u-tilde element of H{sup {beta}}{sup ,0} intersection H{sup 0,{beta}}, {beta}>n/2 are sufficiently small, where H{sup n,m}={l_brace}{phi} element of L{sup 2}: ||<x>{sup m}<i{partial_derivative}{sub x}>{phi}(x)||{sub L{sup 2}}<{infinity}{r_brace}, <x>={radical}(1+x{sup 2}), is a Sobolev weighted space, and that the total mass vector M={integral}u-tilde(x)x{ne}0 is non-zero it is proved that the leading term in the large-time asymptotic expansion of solutions in the critical case is a self-similar solution defined uniquely by the total mass vector M of the initial data.
- OSTI ID:
- 21260476
- Journal Information:
- Sbornik. Mathematics, Vol. 195, Issue 11; Other Information: DOI: 10.1070/SM2004v195n11ABEH000858; Country of input: International Atomic Energy Agency (IAEA); ISSN 1064-5616
- Country of Publication:
- United States
- Language:
- English
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